When is the multiplicity of a weight equal to 1? (Q2639150)

Let G be a complex semisimple Lie algebra and V an irreducible representation corresponding to a highest weight \(\lambda\) and \(K_{\lambda,\mu}\) the multiplicity of an arbitrary weight \(\mu\). The authors describe weights such that \(K_{\lambda,\mu}=1\). Since \(W\cdot \mu \cap P^+\neq \emptyset\) (W Weyl group) it is sufficient to consider the case when \(\mu \in P^+\) where \(P^+\) denotes the semigroup of all dominant weights. The calculation of \(K_{\lambda,\mu}\) is reduced to the case when (\(\lambda\),\(\mu\)) is a primitive pair. A pair (\(\lambda\),\(\mu\)) is said to be primitive if G is simple and \(\lambda - \mu =\sum n_ i\alpha_ i\), \(n_ i>0\). The authors give the list of primitive pairs (\(\lambda\),\(\mu\)) such that \(K_{\lambda,\mu}=1\). In particular, if \(\lambda\) is regular then \(K_{\lambda,\mu}=1\) if and only if \(\lambda\)-\(\mu\) is a linear combination of mutually orthogonal simple roots. There are also other interesting results.